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locus of incenter is a line, starting with non intersecting circles, ext. each

Source: 2009 Oral Moscow Geometry Olympiad grades 8-9 p6

September 19, 2020
geometryincentercircles

Problem Statement

Fixed two circles w1w_1 and w2w_2, \ell one of their external tangent and mm one of their internal tangent . On the line mm, a point XX is chosen, and on the line \ell, points YY and ZZ are constructed so that XYXY and XZXZ touch w1w_1 and w2w_2, respectively, and the triangle XYZXYZ contains circles w1w_1 and w2w_2. Prove that the centers of the circles inscribed in triangles XYZXYZ lie on one line.
(P. Kozhevnikov)